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1

Image of the World on polyhedral maps and globes

Pędzich P.

Polish Cartographical Review

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2016

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Vol. 48, No. 4

197--210

EN

Application of polyhedrons as image surface in cartographic projections has a tradition of more than 200 years. The first maps relying on polyhedrons appeared in the 19th century. One of the first maps which based on an original polyhedral projection using a regular octahedron was constructed by the Californian architect Bernard Cahill in 1909. Other well known polyhedral projections and maps included Buckminster Fuller’s projection and map into icosahedron from 1954 and S. Waterman’s projection into truncated octahedron from 1996, which resulted in the “butterfly” map. Polyhedrons as image surface have the advantage of allowing a continuous image of continents of the Earth with low projection distortion. Such maps can be used for many purposes, such as presentation of tectonic plates or geographic discoveries. The article presents most well known polyhedral maps, describes cartographic projections applied in their preparation, as well as contemporary examples of polyhedral maps. The method of preparation of a polyhedral map and a virtual polyhedral globe is also presented.

2

3D models of regular polyhedrons in: rhinoceros 3D Autocad, 3 Max – possible applications in the teaching of engineering graphics

Nassery F.

Journal Biuletyn of Polish Society for Geometry and Engineering Graphics

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2015

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Vol. 27

37--44

EN

The article presents possible applications in the teaching of engineering graphics of regular polyhedrons 3D modeling with the use of Rhinoceros 3D, AutoCAD and 3ds Max. The three software applications have been compared in terms of their functionalities with respect to the creation of 3D models of regular polyhedrons. It has also been demonstrated how the Rhino can be used for creating projections, plane sections and nets of regular solids on the basis of their 3D models. The article also gives some suggestions on how the above mentioned functionalities of Rhinoceros 3D could be used as teaching tools at technical universities. Firstly, they could help in the visualization and solution of descriptive geometry problems. Secondly, they could serve as tools for creating practice tasks and illustrations to supplement lectures or publications on descriptive geometry or engineering graphics.

PL

W artykule przedstawiono możliwości i sposoby modelowania 3D brył foremnych w programach Rhinoceros 3D oraz AutoCAD i 3ds Max pod kątem wykorzystania w nauczaniu grafiki inżynierskiej. Porównano możliwości poszczególnych programów w tym zakresie. Zademonstrowano tworzenie rzutów, przekrojów zadaną płaszczyzną oraz siatek brył foremnych przy użyciu funkcji programu Rhinoceros na bazie ich trójwymiarowych modeli. Podano również propozycje wykorzystania podanych funkcji programu Rhinoceros 3D na wyższych studiach technicznych jako pomoc przy wizualizacji i rozwiązywaniu zadań z geometrii wykreślnej oraz jako narzędzie do tworzenia tematów, ilustracji do wykładów lub publikacji z dziedziny geometria wykreślna czy też grafika inżynierska.

3

Monge method in creating regular polyhedrons models

Koźniewski E

Journal Biuletyn of Polish Society for Geometry and Engineering Graphics

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2014

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Vol. 26

41--46

EN

Physical face (paper) models of polyhedrons, particularly Platonic solid models, are formed by drawing the nets before them. Other construction requirements makes us the task to run a solid (physical or virtual) model. Beside the solid faces their mutual angles or angles to the selected plane is needed to know. Sometimes other parameters such as a distance parallel faces is good to know. If we choose the geometric way, it does not perform calculations that can be performed using a AutoCAD program. Then turns out to be a very helpful Monge method. The paper contains a proposal to carry out such a task.

PL

Fizyczne ściankowe (np. papierowe) modele wielościanów, w szczególności wielościanów foremnych tworzy się rysując wcześniej na płaszczyźnie ich siatki. Inne wymagania konstrukcyjne stawia nam zadanie wykonania monolitycznego (bryłowego) modelu materialnego za pośrednictwem np. modelu wirtualnego. Obok ścian bryły musimy znać wtedy ich wzajemne kąty nachylenia lub kąty nachylenia do wybranej płaszczyzny lub inne parametry takie jak odległość ścian równoległych. Jeśli wybierzemy drogę geometryczną, to bez wykonywania obliczeń można zrealizować to za pomocą programu AutoCAD. Wówczas bardzo pomocną okazuje się metoda Monge’a. Praca zawiera propozycję zrealizowania takiego zadania.

4

Inverse sequences and absolute co-extensors

Ivanšic I., Rubin R.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 3

243-259

EN

Suppose that K is a CW-complex, X is an inverse sequence of stratifiable spaces, and X = lim X. Using the concept of semi-sequence, we provide a necessary and sufficient condition for X to be an absolute co-extensor for K in terms of the inverse sequence X and without recourse to any specific properties of its limit. To say that X is an absolute co-extensor for K is the same as saying that K is an absolute extensor for X, i.e., that each map ƒ : A → K from a closed subset A of X extends to a map F : X → K. Incase K is & polyhedron |/C|cw (the set \K\ with the weak topology CW), we determine a similar characterization that takes into account the simplicial structure of K.

5

Krasinkiewicz maps from compacta to polyhedra

Matsuhashi E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2006

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Vol. 54, no 2

137-146

EN

We prove that the set of all Krasinkiewicz maps from a compact metric space to a polyhedron (or a 1-dimensional locally connected continuum, or an n-dimensional Menger manifold, n ≥ 1) is a dense Gδ-subset of the space of all maps. We also investigate the existence of surjective Krasinkiewicz maps from continua to polyhedra.

6

O dwóch wielościanach wygenerowanych z ikosaedru

Mirski J. Z.

Materiały Seminarium Geometrii i Grafiki Inżynierskiej

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1999

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z. 7

25--29

EN

In this work the topology of networks is presented of chosen polyhedrons generated by transformations from icosahedron. Thę ideas of the polyhedron and spherical network are introduced interchangeably because of the identity of polyhedron vertices and truss joints of the spherical network. Each of obtained networks is treated as a full one-layer structure. Then two one-layer networks are placed and nodes are properly connected and as the result the full two-layer structure is produced. In the Table I are placed the calculated spherical co-ordinates of nodes, necessary for further computation [4].